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Identify Linear vs. Nonlinear Functions

Students interpret y = mx + b as defining a linear function whose graph is a straight line with constant rate of change. They identify and give examples of nonlinear functions (e.g., y = x²) from tables (non-constant differences) and graphs (non-straight lines), distinguishing them from linear functions.

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Identify Linear vs. Nonlinear Functions

Grade 8 · Functions · question 1 of 3

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20 rungs. A star at the top.

PQRS012345612345678910

Which one of P, Q, R and S has a constant rate of change?

Choose the graph that is a straight line.

How it is taught

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A linear function graphs as one straight line.

Its rate of change stays the same.

A nonlinear graph bends or changes rate.

PQ01234567824681012

Read it: Linear or not linear

A linear function is a straight line: y changes by the same amount for every step in x. Line L rises evenly, so it is linear. Graph K curves, so its steps grow, and it is not linear.

  1. Look at the whole graph: is it one straight line?
  2. If it curves or bends at a corner, it is not linear.
  3. On a table, check that each step in x changes y by the same amount.

A straight line has one rate of change everywhere, so equal steps across give equal steps up.

A curve's steps change as it goes, so no one rate fits it.

A bend is two rates joined at a corner, so a bent graph is not linear either.

Linear means one straight line: the same change in y for every step in x.

Calling any graph that rises linear.

Rising is not enough: K rises too, but it curves, so its rate changes. Only a straight line is linear.

A table gives x = 0, 1, 2, 3 and y = 3, 7, 11, 15. By how much does y change for each step in x? — 4. 7 − 3 = 4, 11 − 7 = 4, 15 − 11 = 4: the same change every step, so the function is linear.

y = 2x vs. y = x²

How do we tell if a function is linear or nonlinear? Let's compare two examples.

First up: y equals two x. Could this be linear?

Let's build a table. We'll pick four x-values: zero, one, two, three.

Multiply each x by two: zero, two, four, six.

Now look at the change in y each step. Plus two, plus two, plus two — a constant rate of change.

Plot the points on a graph. They line up perfectly straight.

Constant differences and a straight line — y equals two x is linear.

Now try y equals x squared, with the same x-values.

Square each x: zero, one, four, nine.

The differences are plus one, plus three, plus five — they keep growing! Not constant.

Plot the points. They curve upward instead of forming a straight line.

Linear or Nonlinear?Compare a table and a graphy = 2xvs.y = x²y = 2xy = x²xyΔy01230246+2+2+2Constant — same Δy every step0149+1+3+5Not constant — Δy keeps growingxy0VERDICTLINEARstraight lineVERDICTNONLINEARcurvedHow to tell them aparty = m x + b → LINEARstraight line · constant Δyy = x² → NONLINEARcurved · Δy keeps changingRULECheck Δyconstant = linear
Grade 8 · Spotting linear and nonlinear from tables and graphs

Rule: y equals m x plus b is linear — straight line, constant differences. Anything else, like y equals x squared, is nonlinear.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

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Divide within 100

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MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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